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If sin θ( 2 sin θ + 3) = 2, 0° < θ < 90°, then what is the value of (sec2 θ + cot2 θ - cos2 θ)? 
1. \(\frac{43}{12}\)
2. \(\frac{13}{3}\)
3. \(\frac{31}{12}\)
4. \(\frac{7}{2}\)

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Correct Answer - Option 1 : \(\frac{43}{12}\)

Given:

 sin θ( 2 sin θ + 3) = 2

Formula used:

Sin 30° = 1/2

Sec 30° = 2/√3

Cos 30° = √3/2

Cot 30° = √3

Calculation:

sin θ( 2 sin θ + 3) = 2

⇒ 2sin2θ + 3sinθ – 2 = 0

⇒ 2sinθ(sinθ + 2) – 1(sinθ + 2)

⇒ sinθ = 1/2

⇒ θ = 30° 

Now,

(sec2 θ + cot2 θ - cos2 θ)

⇒ (2/√3)2 + (√3)2 – (√3/2)2

⇒ (4/3 + 3 – 3/4)

⇒ (16 + 36 – 9)/12

⇒ 43/12

∴ Required value is 43/12

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